A note on Titchmarsh divisor problem under the generalized Riemann hypothesis

Abstract

Let Λ(n) be the von Mangoldt function and τ(n) the divisor function. We focus on the summation T ( x ) = ∑ 1 < n ≤ x Λ ( n ) τ ( n − 1 ) . Under the Riemann hypothesis related to Dirichlet L-functions, it is proved that T ( x ) = x P 1 ( log x ) + O ( x 1 − 𝜃 ) holds for 𝜃<16×10−4. Here P1(t) is a polynomial of degree 1.